Every Moufang loop of odd order with nontrivial commutant has nontrivial center

We get a partial result for Phillips’ problem: does there exist a Moufang loop of odd order with trivial nucleus? First we show that a Moufang loop Q of odd order with nontrivial commutant has nontrivial nucleus, then, by using this result, we prove that the existence of a nontrivial commutant implies the existence of a nontrivial center in Q. Introducing the notion of commutantly nilpotence, we get that the commutantly nilpotence is equivalent to the centrally nilpotence for the Moufang loops of odd order.     

TORSION THEORIES FOR FINITE VON NEUMANN ALGEBRAS

ABSTRACT

The study of modules over a finite von Neumann algebra 𝒜 can be advanced by the use of torsion theories. In this work, some torsion theories for 𝒜 are presented, compared, and studied. In particular, we prove that the torsion theory (T, P) (in which a module is torsion if it is zero-dimensional) is equal to both Lambek and Goldie torsion theories for 𝒜.

Using torsion theories, we describe the injective envelope of a finitely generated projective 𝒜-module and the inverse of the isomorphism K ₀(𝒜) → K ₀ (𝒰), where 𝒰 is the algebra of affiliated operators of 𝒜. Then the formula for computing the capacity of a finitely generated module is obtained. Lastly, we study the behavior of the torsion and torsion-free classes when passing from a subalgebra ? of a finite von Neumann algebra 𝒜 to 𝒜. With these results, we prove that the capacity is invariant under the induction of a ?-module.     

A Generalization of Noncommuting Graph via Automorphisms of a Group

Let G be a group. By using a family 𝒜 of subsets of automorphisms of G, we introduced a simple graph Γ_𝒜(G), which is a generalization of the non-commuting graph. In this paper, we study the combinatorial properties of Γ_𝒜(G).     

On the Bruck-Slaby theorem for commutative Moufang loops

With the help of the relationship between commutative Moufang loops and alternative commutative algebras, we prove, rather easily, the following weakened version of the Bruck-Slaby theorem: a finitely generated commutative Moufang loop is centrally nilpotent. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 275–281, August, 1999.     

Quelques consequences de la locale nilpotence des boucles de moufang commutatives

It is well-known that any finitely generated commutative Moufang loop (CML) is centrally nilpotent and has a finite derived subloop. Consequently such a loop possesses all the classical properties of noethe-rianity: any subloop is finitely generated too, any surjective endomorphism is an automorphism, etc. Besides we prove that, in any CML E(finitely generated or not) the maximal subloops are normal of prime index ; thus the Frattini quotient E_/Φ(E) is an abelian group, sub-direct product of groups of prime order. We shall study also some dual notion of the Frattini subloop, namely the subloop φ^*(E) generated by the minimal normal subloops ; it turns out that φ^*(E) is made up by the products of the prime order central elements.     

Balanced pairs induce recollements

Let 𝒜 be an abelian category with arbitrary (set-indexed) coproducts and exact products. Let (𝒫,?) be a complete balanced pair. Then as in the classical case, we prove that there exists a recollement with the middle term K(𝒜), the homotopy category of 𝒜. In particular, this implies that the relative derived category exists. Two applications are given.     

Two classes of finite groups whose Chermak-Delgado lattice is a chain of length zero

It is an open question in the study of Chermak-Delgado lattices precisely which finite groups G have the property that 𝒞𝒟(G) is a chain of length 0. In this note, we determine two classes of groups with this property. We prove that if G = AB is a finite group, where A and B are abelian subgroups of relatively prime orders with A normal in G, then the Chermak-Delgado lattice of G equals {AC_B(A)}, a strengthening of earlier known results.     

Residual Properties of Free Products

Let 𝒞 be a class of groups. We give sufficient conditions ensuring that a free product of residually 𝒞 groups is again residually 𝒞, and analogous conditions are given for LE-𝒞 groups. As a corollary, we obtain that the class of residually amenable groups and the one of locally embeddable into amenable (LEA) groups are closed under taking free products.

Moreover, we consider the pro-𝒞 topology and we characterize special HNN extensions and amalgamated free products that are residually 𝒞, where 𝒞 is a suitable class of groups. In this way, we describe special HNN extensions and amalgamated free products that are residually amenable.     

Symmetric Elements,Hermitian Forms,and Cyclic Involutions

Let k be a field, K/k be a quadratic separable field extension, and 𝒜 a finite dimensional central simple algebra over K. If k is global or the field of fractions of a two-dimensional excellent henselian local domain with an algebraically closed residue field of characteristic zero and the degree of 𝒜 is odd, we prove that all K/k-involutions on 𝒜 are cyclic.     

Hopf coactions on commutative algebras generated by a quadratically independent comodule

Let 𝒜 be a commutative unital algebra over an algebraically closed field k of characteristic ≠2, whose generators form a finite-dimensional subspace V, with no nontrivial homogeneous quadratic relations. Let 𝒬 be a Hopf algebra that coacts on 𝒜 inner-faithfully, while leaving V invariant. We prove that 𝒬 must be commutative when either: (i) the coaction preserves a non-degenerate bilinear form on V; or (ii) 𝒬 is co-semisimple, finite-dimensional, and char(k) = 0.     

On pure Goldie dimensions

In this paper, we examine the pure Goldie dimension and dual pure Goldie dimension in finitely accessible additive categories. In particular, we show that if A is an object in a finitely accessible additive category 𝒜 that has finite pure Goldie dimension n and finite dual pure Goldie dimension m, then End_𝒜(A) is semilocal and the dual Goldie dimension of End_𝒜(A) is less than or equal to n+m.     

Simple Bol Loops

In this article we investigate the Bol loops and connected with them groups. We prove an analog of the Doro's theorem for Moufang loops and find a criterion for simplicity of Bol loops. One of the main results obtained is the following: If the right multiplication group of a connected finite Bol loop S is a simple group, then S is a Moufang loop.     

Jordan generalized derivations on triangular algebras

Let 𝒜 be a unital algebra and let ? be a unitary 𝒜-bimodule. We consider Jordan generalized derivations mapping from 𝒜 into ?. Our results on unitary algebras are applied to triangular algebras. In particular, we prove that any Jordan generalized derivation of a triangular algebra is a generalized derivation.     

Generalized skew derivations on triangular algebras determined by action on zero products

For a triangular algebra 𝒜 and an automorphism σ of 𝒜, we describe linear maps F,G:𝒜→𝒜 satisfying F(x)y+σ(x)G(y) = 0 whenever x,y∈𝒜 are such that xy = 0. In particular, when 𝒜 is a zero product determined triangular algebra, maps F and G satisfying the above condition are generalized skew derivations of the form F(x) = F(1)x+D(x) and G(x) = σ(x)G(1)+D(x) for all x∈𝒜, where D:𝒜→𝒜 is a skew derivation. When 𝒜 is not zero product determined, we show that there are also nonstandard solutions for maps F and G.     

Nonlinear generalized Lie triple derivation on triangular algebras

Let ? be a commutative ring with identity and let 𝔄 = Tri(𝒜,?,?) be a triangular algebra consisting of unital algebras 𝒜,? over ? and an (𝒜,?)-bimodule ? which is faithful as a left 𝒜-module as well as a right ?-module. In this paper, we prove that under certain assumptions every nonlinear generalized Lie triple derivation G_L:𝔄→𝔄 is of the form G_L = δ+τ, where δ:𝔄→𝔄 is an additive generalized derivation on 𝔄 and τ is a mapping from 𝔄 into its center which annihilates all Lie triple products [[x,y],z].     

Calculus of Functorial Morphisms

We prove that, if F, G: 𝒞 → 𝒟 are two right exact functors between two Grothendieck categories such that they commute with coproducts and U is a generator of 𝒞, then there is a bijection between Nat(F, G) and the centralizer of Hom_𝒟(F(U), G(U)) considered as an Hom_𝒞(U, U)-Hom_𝒞(U, U)-bimodule. We also prove a dual of this result and give applications to Frobenius functors between Grothendieck categories.     

Finite groups with an almost regular automorphism of order four

P. Shumyatsky’s question 11.126 in the “Kourovka Notebook” is answered in the affirmative: it is proved that there exist a constant c and a function of a positive integer argument f(m) such that if a finite group G admits an automorphism ϕ of order 4 having exactly m fixed points, then G has a normal series G ⩾ H ⩽ N such that |G/H| ⩽ f(m), the quotient group H/N is nilpotent of class ⩽ 2, and the subgroup N is nilpotent of class ⩽ c (Thm. 1). As a corollary we show that if a locally finite group G contains an element of order 4 with finite centralizer of order m, then G has the same kind of a series as in Theorem 1. Theorem 1 generalizes Kovács’ theorem on locally finite groups with a regular automorphism of order 4, whereby such groups are center-by-metabelian. Earlier, the first author proved that a finite 2-group with an almost regular automorphism of order 4 is almost center-by-metabelian. The proof of Theorem 1 is based on the authors’ previous works dealing in Lie rings with an almost regular automorphism of order 4. Reduction to nilpotent groups is carried out by using Hall-Higman type theorems. The proof also uses Theorem 2, which is of independent interest, stating that if a finite group S contains a nilpotent subgroup T of class c and index |S: T | = n, then S contains also a characteristic nilpotent subgroup of class ⩽ c whose index is bounded in terms of n and c. Previously, such an assertion has been known for Abelian subgroups, that is, for c = 1. __________ Translated from Algebra i Logika, Vol. 45, No. 5, pp. 575–602, September–October, 2006.     

Diameter of a direct power of a finite group

We present two conjectures concerning the diameter of a direct power of a finite group. The first conjecture states that the diameter of Gⁿ with respect to each generating set is at most n(|G|?rank(G)); and the second one states that there exists a generating set 𝒜, of minimum size, for Gⁿ such that the diameter of Gⁿ with respect to 𝒜 is at most n(|G|?rank(G)). We will establish evidence for each of the above mentioned conjectures.